Switching Linear Systems (SLSs)

• Switching Linear Systems (SLSs) are dynamic systems defined by alternating among a finite set of linear subsystems based on prescribed switching signals.
• Key analyses use Lyapunov methods, graph-theoretic approaches, and automata to assess stability, controllability, and optimal switching laws.
• Applications span continuous and discrete-time models, stochastic switching, and data-driven designs, providing practical insights for control and optimization.

Switching linear systems (SLSs) are dynamical systems whose evolution alternates among finitely many linear modes according to a switching signal. In continuous time, a basic form is

$$\dot x(t)=A_{\sigma(t)}x(t),\qquad x(0)=x_0\in\mathbb{R}^d,$$

while in discrete time one encounters

$$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$

with $\sigma$ piecewise constant in continuous time or sequence-valued in discrete time. The switching signal may be arbitrary, dwell-time constrained, restricted by a graph, generated by a finite-state automaton or a logical dynamic system, induced by a Markov decision process, random, or state dependent. Across these variants, the core problems are asymptotic stability, stabilizability, reachability and observability, switching-law synthesis, identification, abstraction, and model reduction (Kamalov et al., 2022, Kundu et al., 2013, Zhang et al., 2022).

1. State-space models and switching mechanisms

A standard continuous-time SLS consists of a finite family of linear subsystems

$\dot x(t)=A_i x(t),\qquad i\in P=\{1,\dots,N\},$

and a piecewise-constant switching signal $\sigma:[0,\infty)\to P$, right-continuous with left limits, with switching instants $0=\tau_0<\tau_1<\tau_2<\cdots$. In the mode-dependent dwell-time setting, whenever $\sigma(t)=i$ on $[t_k,t_{k+1})$, the sojourn length satisfies

$\tau_i^- \le t_{k+1}-t_k \le \tau_i^+.$

Discrete-time models use the same mode alphabet and switching logic, but the state update occurs at integer times; representative forms are $x(t+1)=A_{\sigma(t)}x(t)$ and $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$0 (Kamalov et al., 2022, Kundu, 2020).

Restricted switching is commonly encoded by a directed graph $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$1. An edge $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$2 means that a transition from mode $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$3 to mode $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$4 is admissible; minimum and maximum dwell times may be imposed simultaneously. A switching signal $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$5 is then admissible when every consecutive mode pair belongs to $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$6 and every holding time lies in the prescribed interval $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$7 (Kundu, 2020, Kundu, 2020). A more general language-theoretic representation uses a finite-state automaton $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$8; a bi-infinite word $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$9 is admissible when there exists a compatible automaton state sequence, and $\sigma$0 is path-complete for a language $\sigma$1 if every $\sigma$2 is admissible (Berger et al., 2018).

Several extensions change the mechanism generating the switching signal rather than the linear modes. In logic dynamic switching, the signal is produced by a discrete logical network

$\sigma$3

and the semi-tensor product (STP) yields an algebraic state-space representation (ASSR) in which logical and real states are merged into a hybrid system (Zhang et al., 2022). In MDP-governed switching, the modes are states of an MDP $\sigma$4, actions determine transition probabilities, and a policy $\sigma$5 induces a DTMC transition matrix $\sigma$6 (Wu et al., 2019). Random switching leads to piecewise-deterministic Markov processes (PDMPs), with the mode process $\sigma$7 jumping according to continuous-time Markov rates $\sigma$8 (Lagasquie, 2016). State-dependent switching uses a partition $\sigma$9 and a switching map $\dot x(t)=A_i x(t),\qquad i\in P=\{1,\dots,N\},$0, so that the active linear mode is selected by the current state (Iwasaki et al., 29 Sep 2025).

2. Stability notions and Lyapunov structures

The basic asymptotic-stability notion for continuous-time SLSs is global asymptotic stability under arbitrary switching: for every measurable $\dot x(t)=A_i x(t),\qquad i\in P=\{1,\dots,N\},$1 and every initial state $\dot x(t)=A_i x(t),\qquad i\in P=\{1,\dots,N\},$2, the trajectory satisfies $\dot x(t)=A_i x(t),\qquad i\in P=\{1,\dots,N\},$3 as $\dot x(t)=A_i x(t),\qquad i\in P=\{1,\dots,N\},$4. Under dwell-time constraints, one instead asks that $\dot x(t)=A_i x(t),\qquad i\in P=\{1,\dots,N\},$5 for every switching signal satisfying the admissibility restrictions. Classical characterizations include existence of a common quadratic or polyhedral Lyapunov function and negativity of the joint Lyapunov exponent (Kamalov et al., 2022).

A central technique is the use of multiple Lyapunov-like functions. For each subsystem $\dot x(t)=A_i x(t),\qquad i\in P=\{1,\dots,N\},$6, one chooses

$\dot x(t)=A_i x(t),\qquad i\in P=\{1,\dots,N\},$7

together with a scalar $\dot x(t)=A_i x(t),\qquad i\in P=\{1,\dots,N\},$8 such that along mode-$\dot x(t)=A_i x(t),\qquad i\in P=\{1,\dots,N\},$9 trajectories,

$\sigma:[0,\infty)\to P$0

in continuous time, or

$\sigma:[0,\infty)\to P$1

in discrete time. The sign of $\sigma:[0,\infty)\to P$2 encodes the intrinsic mode type: $\sigma:[0,\infty)\to P$3 for asymptotically stable continuous-time modes, $\sigma:[0,\infty)\to P$4 for marginally stable modes, and $\sigma:[0,\infty)\to P$5 for unstable modes; in discrete time, $\sigma:[0,\infty)\to P$6, $\sigma:[0,\infty)\to P$7, and $\sigma:[0,\infty)\to P$8 play the analogous roles. At a switch $\sigma:[0,\infty)\to P$9, mode changes are compared through

$0=\tau_0<\tau_1<\tau_2<\cdots$0

which quantifies the jump cost induced by changing Lyapunov coordinates (Kundu et al., 2013, Kundu et al., 2013).

This framework yields asymptotic criteria in terms of long-run switching statistics. For continuous time, Kundu–Chatterjee introduce the $0=\tau_0<\tau_1<\tau_2<\cdots$1-frequency of switching $0=\tau_0<\tau_1<\tau_2<\cdots$2, transition densities $0=\tau_0<\tau_1<\tau_2<\cdots$3, and activation fractions $0=\tau_0<\tau_1<\tau_2<\cdots$4, together with their $0=\tau_0<\tau_1<\tau_2<\cdots$5 or $0=\tau_0<\tau_1<\tau_2<\cdots$6 asymptotic values. Their stabilizing-switching theorem requires

$0=\tau_0<\tau_1<\tau_2<\cdots$7

and

$0=\tau_0<\tau_1<\tau_2<\cdots$8

which states that the average growth due to switching must remain strictly below the net decay from stable modes minus growth from unstable modes (Kundu et al., 2013). A discrete-time analogue uses the switching frequency $0=\tau_0<\tau_1<\tau_2<\cdots$9, edge-transition counts, and activation times $\sigma(t)=i$0, together with a ratio condition $\sigma(t)=i$1 for a repeated closed walk on the admissibility graph (Kundu et al., 2013).

For continuous-time switching stabilization, four notions are proved equivalent: measurable switching stabilizability, feedback stabilizability in Filippov sense, feedback stabilizability in sample-and-hold sense with bounded sampling rate, and exponentially discrete switching stabilizability. Their necessary-and-sufficient condition is the existence of a pointwise-minimum piecewise-quadratic control-Lyapunov function (pm-PQCLF),

$\sigma(t)=i$2

with a uniform decrease condition

$\sigma(t)=i$3

This result unifies open-loop, feedback, Filippov, sample-and-hold, and discrete-time stabilization viewpoints (Lu et al., 2015).

Not all asymptotic analyses are stability analyses in the strict sense. Path-complete $\sigma(t)=i$4-dominance studies discrete-time switching systems through contraction of quadratic $\sigma(t)=i$5-cones $\sigma(t)=i$6, with inertia $\sigma(t)=i$7, using LMIs of the form

$\sigma(t)=i$8

Path-dominant systems admit a dominated splitting $\sigma(t)=i$9 with $[t_k,t_{k+1})$0, and the $[t_k,t_{k+1})$1-component decays exponentially relative to the $[t_k,t_{k+1})$2-component. This yields a low-dimensional dominant behavior rather than merely asymptotic convergence to the origin (Berger et al., 2018).

3. Dwell times, destabilization, randomness, and communication constraints

Dwell-time constraints can be analyzed mode by mode. For a single Hurwitz matrix $[t_k,t_{k+1})$3, Protasov and Kamalov define the symmetrized convex hull of a trajectory segment by

$[t_k,t_{k+1})$4

and call $[t_k,t_{k+1})$5 a cut tail point of $[t_k,t_{k+1})$6 if for every $[t_k,t_{k+1})$7, one has $[t_k,t_{k+1})$8. The set of cut tail points is a closed half-line $[t_k,t_{k+1})$9, and $\tau_i^- \le t_{k+1}-t_k \le \tau_i^+.$0 is characterized by an extremal quasipolynomial inequality

$\tau_i^- \le t_{k+1}-t_k \le \tau_i^+.$1

In the real-spectrum case, the extremal solution is a Chebyshev-type exponential polynomial with an alternance of $\tau_i^- \le t_{k+1}-t_k \le \tau_i^+.$2 points (Kamalov et al., 2022).

The practical consequence is a dwell-time-selection rule. Suppose a switching system has mode-$\tau_i^- \le t_{k+1}-t_k \le \tau_i^+.$3 lower dwell time $\tau_i^- \le t_{k+1}-t_k \le \tau_i^+.$4. After computing $\tau_i^- \le t_{k+1}-t_k \le \tau_i^+.$5, one sets

$\tau_i^- \le t_{k+1}-t_k \le \tau_i^+.$6

The First fundamental theorem states that if the system is stable under the additional restriction that every mode-$\tau_i^- \le t_{k+1}-t_k \le \tau_i^+.$7 interval is at most $\tau_i^- \le t_{k+1}-t_k \le \tau_i^+.$8, then it is stable without any upper bound. In the formulation given in the paper, the worst-case growth of a stable switching linear system always arises from switching signals whose mode-$\tau_i^- \le t_{k+1}-t_k \le \tau_i^+.$9 intervals do not exceed $x(t+1)=A_{\sigma(t)}x(t)$0 (Kamalov et al., 2022).

A recurring misconception is that the negation of a sufficient stability condition should imply instability. The asymptotic destabilizing-signal theory shows that this is false. For the continuous-time system $x(t+1)=A_{\sigma(t)}x(t)$1, the destabilizing criterion is expressed in terms of the asymptotic switching frequency $x(t+1)=A_{\sigma(t)}x(t)$2, transition frequencies $x(t+1)=A_{\sigma(t)}x(t)$3, and activation fractions $x(t+1)=A_{\sigma(t)}x(t)$4, together with lower-bound Lyapunov constants $x(t+1)=A_{\sigma(t)}x(t)$5 and $x(t+1)=A_{\sigma(t)}x(t)$6. If

$x(t+1)=A_{\sigma(t)}x(t)$7

then $x(t+1)=A_{\sigma(t)}x(t)$8 for every nonzero initial condition. At the same time, the destabilizing class is a strict subset of the complement of the stabilizing class, which identifies a gap between asymptotic characterizations of stabilizing and destabilizing switching signals (Kundu, 2018).

Another common misconception is that stable modes imply stable switching. A planar PDMP constructed from two Hurwitz matrices shows that, even if the two systems are stable, it is possible to obtain a blow up if one chooses the switching rates wisely. The almost-sure Lyapunov exponent

$x(t+1)=A_{\sigma(t)}x(t)$9

changes sign with the switching rates: for each fixed $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$00, as $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$01 or $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$02, $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$03, but for every fixed $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$04 and $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$05, one can choose $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$06 small enough and $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$07 large enough so that $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$08 (Lagasquie, 2016).

Communication constraints add a further layer of restriction. For continuous-time switched linear systems with unknown switching signal, arbitrary switching is in general not stabilizable with a finite data rate; the one-dimensional example $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$09, $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$10, $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$11, $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$12 already exhibits impossibility. Under an Average Dwell Time (ADT) assumption

$$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$13

and a classical stabilizability assumption, there exists a coder-controller with finite averaged data-rate $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$14 that yields exponential decay $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$15. The result shows that classical stabilizability plus a mild ADT guarantee remains sufficient for stabilization over a finite-capacity channel (Berger et al., 2020).

4. Structured languages, logic-generated switching, and formal synthesis

When switching is constrained by a formal language, the automaton itself becomes part of the system description. In path-complete analysis, the allowed words are those admitted by a finite-state automaton $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$16, and the switching system is $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$17-dominant with respect to $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$18 if there is a family of quadratic cones $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$19 contracted along every automaton transition. In strongly connected or path-complete automata, the inertia constraint can be dropped from the SDP because every $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$20 automatically acquires the same inertia $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$21 (Berger et al., 2018).

Logical dynamic switching provides a different formalization. The switching signal is generated by a discrete logical network with structure matrices $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$22 and $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$23, and the pure SLS is rewritten through the semi-tensor product as

$$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$24

By merging logical and continuous states into $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$25, one obtains a hybrid ASSR

$$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$26

Within this representation, reachability, controllability, observability, and reconstructibility are characterized by Kalman-type rank conditions and unions-of-images conditions over admissible logical-input sequences. The same framework gives necessary and sufficient conditions for realization of fixed-operating-time switching and finite reference switching (Zhang et al., 2022).

If the switching law is controllable through an MDP policy, policy synthesis becomes a stability-constrained optimization problem. Given a policy $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$27, the switched linear system becomes a Markov jump linear system (MJLS), and stability is certified by mode-dependent Lyapunov matrices $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$28 satisfying

$$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$29

When the transition matrix $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$30 depends on the policy variables, the policy-synthesis constraints become bilinear matrix inequalities (BMIs). Two convex alternatives are given in the literature: a scalar-Lyapunov SDP relaxation and a coordinate-descent scheme alternating between an SDP in $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$31 and an SDP in $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$32. The BMI approach finds many feasible policies when $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$33 are small; the SDP relaxation is fastest but very conservative; coordinate descent provides a good trade-off in the reported examples (Wu et al., 2019).

Formal verification and switching-law synthesis can also be carried out after finite abstraction. For discrete-time SLSs with stable subsystems, a finite bisimulation quotient can be constructed on a bounded subset of the state space using sublevel sets of a polyhedral Lyapunov function $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$34. The algorithm slices the state space by nested sublevel polytopes $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$35, computes predecessor sets $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$36, and refines a partition until a bisimulation relation is obtained. The quotient can then be used for synthesis of the switching law and system verification with respect to syntactically co-safe Linear Temporal Logic formulas over observed polytopic subsets (Gol et al., 2012).

5. Optimization, graph-theoretic synthesis, and data-driven design

Finite-horizon optimization over switching sequences is already nontrivial for purely linear modes. In the discrete-time problem

$$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$37

one seeks a length-$$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$38 sequence maximizing a convex terminal cost $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$39. The exact dynamic-programming algorithm propagates only the extreme points $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$40 of the convex hull $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$41, because maximizing a convex function over a convex hull reduces to evaluating the extreme points. Polynomial-time solvability is obtained when $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$42 has the oligo-vertex property, meaning $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$43 for all large $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$44. Sufficient conditions include commuting families, sets in which at most one matrix has rank $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$45, pairs of $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$46 matrices sharing a real eigenvector, and pairs of $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$47 binary or right-stochastic matrices (Wu et al., 2018).

Restricted-switching stabilizability can be synthesized directly on a graph. In one approach for discrete-time SLSs with admissible switch set $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$48 and dwell bounds $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$49, subsystem traces are used to build data matrices $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$50, from which feasible pairs $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$51 are computed by LMIs of Park–Ikeda type. These pairs define vertex weights $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$52 and edge weights $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$53, and stability is guaranteed by a contractive cycle

$$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$54

Equivalently, one searches for a negative cycle in a weighted directed graph; the resulting periodic walk yields an admissible periodic switching logic $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$55 that is globally asymptotically stable (Kundu, 2020).

A different graph-theoretic design appears when all discrete-time subsystems are unstable and stabilization must be achieved purely through restricted switching. If there exist two modes $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$56 and dwell times $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$57 such that $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$58 is Schur-stable, then sufficient conditions can be stated in terms of admissibility-graph paths and small commutator norms

$$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$59

The resulting scalar inequalities yield periodic or non-periodic stabilizing switching signals under restricted switching (Kundu, 2020).

Data-driven stability certification for constrained SLSs can also be stated probabilistically. For an unknown discrete-time system $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$60 whose switching words are constrained by a strongly connected automaton $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$61, one samples length-$$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$62 trajectories, solves a sampled quadratic Lyapunov SDP, and obtains with confidence $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$63 an upper bound

$$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$64

for the constrained joint spectral radius. The entropy

$$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$65

bounds the number of samples needed in order to reach a pre-specified accuracy; smaller $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$66 means fewer samples (Banse et al., 2022).

State-dependent SLS identification has recently been formulated as a convex-optimization hierarchy. The joint estimation of mode assignments and mode dynamics begins as a mixed-integer program over one-hot mode indicators $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$67, with $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$68. Two relaxations are used: a moment/Shor SDP relaxation and a simplex $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$69 relaxation. Building on these, a bilevel convex optimization framework alternates between mode assignment and dynamics estimation, and margin-based polynomial classifiers recover switching boundaries. On a switching damped oscillator, both relaxations recover the dynamics with $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$70, mode-accuracy and mIoU both $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$71, and rollout RMSE $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$72; the LP relaxation is reported to be an order-of-magnitude faster than the SDP (Iwasaki et al., 29 Sep 2025).

6. Identification, realization, reduction, and related complexity questions

When the discrete state and continuous state are both unknown, switched-linear system identification requires separating mode estimation from linear realization. One observer-based route transforms a switched state-space model into a switched auto-regressive with exogenous input (SARX) model through a deadbeat observer. If the minimum dwell time satisfies $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$73, then a piecewise-constant observer gain can make the observer nilpotent with $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$74, and in fact $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$75 on sufficiently long intervals. The resulting finite-memory relation

$$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$76

is used in a block-sparse optimization problem; a convex relaxation gives the block basis pursuit denoising (BBPDN) algorithm, long segments are clustered to recover the discrete states, and short segments are handled by MOESP variants. In the reported three-mode example, BBPDN solved via CVX with $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$77, and MOESP then recovered $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$78 exactly except $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$79–$$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$80 samples (Bencherki et al., 2021).

For MIMO systems, realization from Markov parameters has been organized into four stages. First, a topologically equivalent linear time-varying realization is obtained from block Hankel matrices $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$81. Second, zero sets of Hankel differences isolate stationary subintervals on which the realized system has an LTI pulse response matching that of the original SLS. Third, the switching sequence is estimated by forward/backward corrections and Markov-parameter matching. Fourth, basis transformations align all recovered submodels into a common state basis. Under common MacMillan degree and dwell-time conditions, the method recovers all submodels up to similarity transformations and the switching sequence exactly in the noiseless example (Bencherki et al., 2021).

Large-scale switched models motivate reduction rather than identification. For linear systems with low-rank switching, where $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$82 with $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$83, the system can be replaced by an envelope system, an LTI MIMO model with extended inputs and outputs. A feedback law depending on the active mode reproduces the switched output exactly, after which standard Petrov–Galerkin model order reduction may be applied. The reduced envelope system induces a reduced switched model, an $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$84-type output error bound is available under a small-gain condition, and quadratic Lyapunov stability is preserved by choosing $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$85 when the original switched system is quadratic Lyapunov stable (Schulze et al., 2018).

Some SLSs arise from communication architecture rather than explicit mode design. Discrete-time linear systems with switching propagation delays can be modeled as arbitrary-switching systems by augmenting the state with delay-pipeline variables, leading to

$$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$86

In this representation, robust stability is equivalent to the joint spectral radius $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$87. Approximate stability can be decided in finite time, but asymptotic stability is NP-hard even in the case $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$88 and nonnegative rational entries, and deciding boundedness $$x(t+1)=A_{\sigma(t)}x(t),\qquad x(t+1)=A_{\sigma(t)}x(t)+B_{\sigma(t)}u(t),$$89 is Turing-undecidable (Jungers et al., 2014).

Taken together, these results suggest a broad research picture. SLS theory spans continuous-time and discrete-time models, open-loop and feedback switching, arbitrary and constrained languages, deterministic and stochastic mechanisms, fully known and data-driven settings, and exact as well as reduced-order representations. The available analyses show both unifying structure—Lyapunov functions, graph walks, automata, SDP or LMI formulations, convex hulls, and Markov parameters—and intrinsic limits, including strict gaps between asymptotic stability and instability criteria, finite-data-rate impossibility under arbitrary switching, and hardness of stability analysis in structured subclasses (Kundu, 2018, Berger et al., 2020, Jungers et al., 2014).